Data605_Discussion12
Aaron Zalki
4/18/2020
##
## Welcome to CUNY DATA606 Statistics and Probability for Data Analytics
## This package is designed to support this course. The text book used
## is OpenIntro Statistics, 3rd Edition. You can read this by typing
## vignette('os3') or visit www.OpenIntro.org.
##
## The getLabs() function will return a list of the labs available.
##
## The demo(package='DATA606') will list the demos that are available.The movie Moneyball focuses on the “quest for the secret of success in baseball”. It follows a low-budget team, the Oakland Athletics, who believed that underused statistics, such as a player’s ability to get on base, better predict the ability to score runs than typical statistics like home runs, RBIs (runs batted in), and batting average.
2011 Major League Baseball Season Data For All 30 Teams
This is the same data used in Lab 8
## X team runs at_bats hits homeruns bat_avg strikeouts
## 1 1 Texas Rangers 855 5659 1599 210 0.283 930
## 2 2 Boston Red Sox 875 5710 1600 203 0.280 1108
## 3 3 Detroit Tigers 787 5563 1540 169 0.277 1143
## 4 4 Kansas City Royals 730 5672 1560 129 0.275 1006
## 5 5 St. Louis Cardinals 762 5532 1513 162 0.273 978
## 6 6 New York Mets 718 5600 1477 108 0.264 1085
## 7 7 New York Yankees 867 5518 1452 222 0.263 1138
## 8 8 Milwaukee Brewers 721 5447 1422 185 0.261 1083
## 9 9 Colorado Rockies 735 5544 1429 163 0.258 1201
## 10 10 Houston Astros 615 5598 1442 95 0.258 1164
## 11 11 Baltimore Orioles 708 5585 1434 191 0.257 1120
## 12 12 Los Angeles Dodgers 644 5436 1395 117 0.257 1087
## 13 13 Chicago Cubs 654 5549 1423 148 0.256 1202
## 14 14 Cincinnati Reds 735 5612 1438 183 0.256 1250
## 15 15 Los Angeles Angels 667 5513 1394 155 0.253 1086
## 16 16 Philadelphia Phillies 713 5579 1409 153 0.253 1024
## 17 17 Chicago White Sox 654 5502 1387 154 0.252 989
## 18 18 Cleveland Indians 704 5509 1380 154 0.250 1269
## 19 19 Arizona Diamondbacks 731 5421 1357 172 0.250 1249
## 20 20 Toronto Blue Jays 743 5559 1384 186 0.249 1184
## 21 21 Minnesota Twins 619 5487 1357 103 0.247 1048
## 22 22 Florida Marlins 625 5508 1358 149 0.247 1244
## 23 23 Pittsburgh Pirates 610 5421 1325 107 0.244 1308
## 24 24 Oakland Athletics 645 5452 1330 114 0.244 1094
## 25 25 Tampa Bay Rays 707 5436 1324 172 0.244 1193
## 26 26 Atlanta Braves 641 5528 1345 173 0.243 1260
## 27 27 Washington Nationals 624 5441 1319 154 0.242 1323
## 28 28 San Francisco Giants 570 5486 1327 121 0.242 1122
## 29 29 San Diego Padres 593 5417 1284 91 0.237 1320
## 30 30 Seattle Mariners 556 5421 1263 109 0.233 1280
## stolen_bases wins new_onbase new_slug new_obs
## 1 143 96 0.340 0.460 0.800
## 2 102 90 0.349 0.461 0.810
## 3 49 95 0.340 0.434 0.773
## 4 153 71 0.329 0.415 0.744
## 5 57 90 0.341 0.425 0.766
## 6 130 77 0.335 0.391 0.725
## 7 147 97 0.343 0.444 0.788
## 8 94 96 0.325 0.425 0.750
## 9 118 73 0.329 0.410 0.739
## 10 118 56 0.311 0.374 0.684
## 11 81 69 0.316 0.413 0.729
## 12 126 82 0.322 0.375 0.697
## 13 69 71 0.314 0.401 0.715
## 14 97 79 0.326 0.408 0.734
## 15 135 86 0.313 0.402 0.714
## 16 96 102 0.323 0.395 0.717
## 17 81 79 0.319 0.388 0.706
## 18 89 80 0.317 0.396 0.714
## 19 133 94 0.322 0.413 0.736
## 20 131 81 0.317 0.413 0.730
## 21 92 63 0.306 0.360 0.666
## 22 95 72 0.318 0.388 0.706
## 23 108 72 0.309 0.368 0.676
## 24 117 74 0.311 0.369 0.680
## 25 155 91 0.322 0.402 0.724
## 26 77 89 0.308 0.387 0.695
## 27 106 80 0.309 0.383 0.691
## 28 85 86 0.303 0.368 0.671
## 29 170 71 0.305 0.349 0.653
## 30 125 67 0.292 0.348 0.640I am interested in creating a linear model to predict the number of wins a team achieves in a season based on how many homeruns the team scores.
homeruns as the predictor
wins as the output
Plot Data
#x axis is home runs, y axis is wins
homerun_wins <- ggplot(df, aes(homeruns, wins))
homerun_wins + geom_point()
The plot concludes an approximately linear relationship with a 66% correlation strength as seen below. The two variables are positively correlated meaning the more home runs the more wins. The strength is moderately strong because there is variation in the data.
## [1] 0.660614Sum of Squared Residuals
Similar to how we can use the mean and standard deviation to summarize a single variable, we can summarize the relationship between homerunsand wins by finding the line that best follows their association.
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 48.8140 0.2119
##
## Sum of Squares: 2129.785There are 30 residuals shown in blue, one for each of the 30 MLB teams. Residuals are the difference between the observed values and the values predicted by the line:
\[ e_i = y_i - \hat{y}_i \]
The most common way to do linear regression is to select the line that minimizes the sum of squared residuals. To visualize the squared residuals, I will rerun the plot_ss command and add the argument showSquares = TRUE.
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 48.8140 0.2119
##
## Sum of Squares: 2129.785The output from the plot_ss function provides the slope and intercept of the line as well as the sum of squares. I can also capture this info using the lm function in R (Textbook Page 19) to fit the linear model or regression line.
The Linear Model
The output of lm is an object that contains all of the information we need about the linear model that was just fit. We can access this information using the summary function.
##
## Call:
## lm(formula = wins ~ homeruns, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -20.2874 -6.7083 0.7708 5.6292 20.7649
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 48.81397 7.08635 6.888 1.74e-07 ***
## homeruns 0.21190 0.04551 4.656 7.09e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.721 on 28 degrees of freedom
## Multiple R-squared: 0.4364, Adjusted R-squared: 0.4163
## F-statistic: 21.68 on 1 and 28 DF, p-value: 7.094e-05The “Coefficients” table shown is key; its first column displays the linear model’s y-intercept and the coefficient of homeruns. With this table, we can write down the least squares regression line for the linear model:
\[ \hat{y} = 48.81397 + 0.21190 * {home runs} \]
The \(R^2\) value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 43.6% of the variability in wins is explained by home runs.
Prediction and prediction errors
Let’s create a scatterplot with the least squares line laid on top.
The function abline plots a line based on its slope and intercept. Here, we used a shortcut by providing the model linear_model, which contains both parameter estimates. This line can be used to predict \(y\) at any value of \(x\). When predictions are made for values of \(x\) that are beyond the range of the observed data, it is referred to as extrapolation and is not usually recommended. However, predictions made within the range of the data are more reliable. They’re also used to compute the residuals.
Model diagnostics
To assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.
Linearity: We checked if the relationship between homeruns and wins is linear using a scatterplot. We should also verify this condition with a plot of the residuals vs. homeruns.
plot(linear_model$residuals ~ df$homeruns)
abline(h = 0, lty = 3) # adds a horizontal dashed line at y = 0
There is no obvious pattern in the residual plot
Nearly normal residuals: To check this condition, we can look at a histogram or a normal probability plot of the residuals.
qqnorm(linear_model$residuals)
qqline(linear_model$residuals) # adds diagonal line to the normal prob plot
The residuals appear to be constant and normal, therefore we can assume the constant variability condition is met.